A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, 2
Abstract
Description
Let $ρ(x)=x-[x]$, $χ=χ_{(0,1)}$. In $L_2(0,\infty)$ consider the subspace $\B$ generated by $\{ρ_a|a\geq1\}$ where $ρ_a(x):=ρ(\frac{1}{ax})$. By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the statement $χ\in\bar{\B}$. For some time it has been conjectured, and proved in the first version of this paper, posted in arXiv:math.NT/0202141 v2, that the Riemann hypothesis is equivalent to the stronger statement that $χ\in\bar{\Bnat}$ where $\Bnat$ is the much smaller subspace generated by $\{ρ_a|a\in\Nat\}$. This second version differs from the first in showing that under the Riemann hypothesis for some constant $c>0$ the distance between $χ$ and $-\sum_{a=1}^nμ(a)e^{-c\frac{\log a}{\log\log n}}ρ_a$ is of order $(\log\log n)^{-1/3}$.
9 pages
9 pages